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\title{Permutation}
\author{Space and Time Inc}
\date{February 2025}

\begin{document}
\maketitle

\noindent Let $A=(a_{ij})$ be a table. We need to prove that $R=(r_{ij})$ is a permuted version of $A$. \\

\section{Summary}
\begin{itemize}
    \item Plan values: $i$ and number of columns in $A$
    \item Inputs: $A$
    \item Outputs: $R$
    \item Hints: $c$, $d$, $c^\ast$, $d^\ast$.
\end{itemize}

\section{Details}
We set 

\begin{align*}
    \hat{c} &\equiv \sum_j a_j \beta^j &\hat{d} &\equiv \sum_j r_ j \beta^j
\end{align*}

The $3$ constraints are:

\begin{align*}
    c^\ast \cdot (\alpha + \hat{c}) &\equiv \chi_{[0,n)}\\
    d^\ast \cdot (\alpha + \hat{d}) &\equiv \chi_{[0,n)}\\
    c^\ast &\overset{\sum}{=} d^\ast
\end{align*}
\end{document}
